Bank Math

Question: A mixture of milk and water has a total volume of 42 liters, where milk and water are mixed in the ratio 3 : 4. How many liters of milk must be added so that the quantities of milk and water become equal?

Solution:
The ratio of milk to water is 3 : 4
Total portion = 3 + 4 = 7

Quantity of milk = 42 × (3/7) = 18 liters.
Quantity of water = 42 × (4/7) = 24 liters.

Let,
Quantity of milk to be added = x liters

According to the question,
(18 + x) : 24 = 1 : 1
⇒ (18 + x)/24 = 1/1
⇒ 18 + x = 24
⇒ x = 24 - 18
⇒ x = 6

∴ Quantity of milk to be added = 6 liters

Question: What is the angle between the hour and minute hand of a clock when it is 3 : 40 pm?
​
​Solution:
​3 ঘণ্টা 40 মিনিট = 3 + (40/60) ঘণ্টা
= 11/3 ঘণ্টা

​ঘণ্টার কাঁটা 12 ঘণ্টায় 360° ঘুরে
​∴ 1 ঘণ্টায় ঘুরে 360/12 = 30°
​∴​ 11/3 ঘণ্টায় ঘুরে (30 × 11/3) = 110°

​আবার,
​মিনিটের কাঁটা প্রতি মিনিটে 6° করে ঘুরে
​∴ 40 মিনিটে ঘুরে (6 × 40) = 240°

সুতরাং, ঘড়ির কাঁটা দুটির মধ্যবর্তী কোণ​ = |240 - 110| = 130°

Question: In how many ways can 6 books be selected from 10 books such that 2 particular books are always excluded?

Solution:
Since 2 specific books must always be excluded, we are effectively choosing from the remaining books.
Remaining books = 10 - 2 = 8 books

Now, we need to select 6 books from these 8 books.
The number of ways to do so is given by the combination formula = ⁸C₆
= 8!/(6! × (8 - 6)!)
= 8!/(6! × 2!)
= (8 × 7 × 6!)/(6! × 2 × 1)
= (8 × 7)/2
= 56/2
= 28

So, the total number of ways 28. 

মনে করি, AB = h
খুঁটিটি ভূমির সাথে 60° কোণ তৈরি করে BC = √3 মিটার ছায়া তৈরি করে
তাহলে খুঁটির উচ্চতা h = ?
প্রশ্নমতে, tan60° = AB / BC
⇒ √3 = h/√3
∴ h = √3.√3 = 3 মিটার 

Question:

Solution:

Question: Find the compound interest on Tk. 2000 at the rate of 20% per annum for 1.5 years. When interest is compounded half-yearly.

Solution: 
Principal P = 2000
Rate, R = 20%
Time, T = 1.5 years

Now,
Compound interest for half-yearly:
A = P{1 + R/(2×100)}^2T
= 2000{1 + (20/200)}^2×1.5
= 2000 × {1 + (1/10)}³
= 2000 × (11/10)³ 
= 2000 × (1331/1000)
= 2662

Compound Interest = A – P
= 2662 – 2000 = 662 Tk

The word,'APPLE' contans 5 letters, 1A, 2P, 1L and 1E.
∴ Required number of ways = 5!/2!
= 60

According to the question,
(x + y)/(x - y) = 4
⇒ x + y = 4x - 4y
⇒ 4x - x = 4y + y
⇒ 3x = 5y
⇒ x/y = 5/3
⇒ x²/y² = 25/9
Now,
⇒ x² + y²/x² - y²
= {(x² /y²) + 1}/{(x²/y²) - 1}
= {(25/9) +1}/{(25/9 - 1}
(34/9) × (9/16)
= 17/8

Question: A cyclist covers 15 rounds of a circular track of 400 meters every day. The time taken by the cyclist for three consecutive days are 75, 85, and 80 minutes respectively. On an average, what is the speed of the cyclist in meters/minute?

সমাধান:
মোট অতিক্রান্ত দূরত্ব (৩ দিনে) = (দিনের সংখ্যা × প্রতি দিনের রাউন্ড × ট্র্যাকের দৈর্ঘ্য)
= (3 × 15 × 400) মিটার
= 18000 মিটার।

মোট সময় লেগেছে = (75 + 85 + 80) মিনিট
= 240 মিনিট।

গড় গতিবেগ = মোট দূরত্ব / মোট সময়
= 18000 / 240 মিটার/মিনিট
= 75 মিটার/মিনিট।

∴ সাইকেল আরোহীর গড় গতিবেগ হলো 75 মিটার/মিনিট।

Question: Father is aged three times more than his son Ronit. After 8 years, he would be two and a half times of Ronit's age. After further 8 years, how many times would he be of Ronit's age?

Solution:
Let Ronit's present age be x years.
Then, father's present age =(x + 3x) years
= 4x years

Now
4x + 8 = (5/2)(x + 8)
8x + 16 = 5x + 40
8x - 5x = 40 - 16
3x = 24
x = 8

Further 8 years will means 16 years from the current age.
After 16 years from present, Ronit's age be (8 + 16) = 24 years
After 16 years from present, father's age be (8 × 4 + 16) = 48 years

After 16 years from present the ratio of their ages will be 
His father's age/ Ronit's age = 48/24 
His father's age/ Ronit's age =2
His father's age = 2 × Ronit age

Question: Two pipes, Pipe X and Pipe Y, can fill a tank in 20 hours and 30 hours, respectively. If both pipes are opened together, after how many hours should Pipe Y be closed so that the tank is completely filled in 15 hours?

Solution:
ধরি, ট্যাঙ্কটির ধারণক্ষমতা হলো LCM (20, 30) = 60 ইউনিট।
পাইপ X-এর কর্মদক্ষতা = 60 / 20 = 3 ইউনিট/ঘণ্টা।
পাইপ Y-এর কর্মদক্ষতা = 60 / 30 = 2 ইউনিট/ঘণ্টা।
পাইপ X এবং Y-এর মিলিত কর্মদক্ষতা = 3 + 2 = 5 ইউনিট/ঘণ্টা।

ধরি, পাইপ X এবং পাইপ Y একত্রে চলে n ঘণ্টা।
∴ পাইপ Y বন্ধ করার পর পাইপ X একা (15 - n) ঘণ্টা চলে।

প্রশ্নানুসারে,
5n + 3(15 - n) = 60
⇒ 5n + 45 - 3n = 60
⇒ 2n + 45 = 60
⇒ 2n = 60 - 45
⇒ 2n = 15
⇒ n = 15/2
⇒ n = 7.5

∴ পাইপ Y-কে 7.5 ঘণ্টা পর বন্ধ করা উচিত।

Question: If the heights of two cones are in the ratio 7 : 3 and their diameters are in the ratio 6 : 7, what is the ratio of their volumes?

Solution: 
Let the heights of two cones be 7x and 3x, and their diameters be 6y and 7y, respectively
∴ Volume of first cone = (1/3π) × (6y/2)² × 7x
And volume of second cone = (1/3π) × (7y/2)² × 3x 

Then,
Ratio of volume,  

Question: Find the greatest value of sin⁶A + cos⁶A.

Solution:
We know,
sin²A + cos²A = 1
⇒ (sin²A + cos²A)³ = 1³
⇒ (sin²A)³ + (cos²A)³ + 3 sin²A cos²A (sin²A + cos²A) = 1
⇒ sin⁶A + cos⁶A + 3 sin²A cos²A (1) = 1
⇒ sin⁶A + cos⁶A = 1 - 3 sin²A cos²A
⇒ sin⁶A + cos⁶A = 1 - 3 sin²90° cos²90°
[প্রদত্ত রাশির সর্বোচ্চ মান পেতে হলে, 1 থেকে যে রাশিটি বিয়োগ করা হচ্ছে, সেই 3sin²Acos²A রাশিটির সর্বনিম্ন মান হতে হবে। এটি হয় যখন A = 0° অথবা A = 90°]
⇒ sin⁶A + cos⁶A = 1 - 3(1² × 0²)
⇒ sin⁶A + cos⁶A = 1 - 0
∴ sin⁶A + cos⁶A = 1

Question: Two ingoing pipes can fill a tank in 6 hours and 8 hours, respectively. An outgoing pipe is attached to these two pipes and thus the tank was filled in 4 hours. In 48 hours, the outgoing pipe alone can empty how many tanks?

Solution:
Let the outgoing pipe take P hours to empty the tank.

So, in 1 hour, total fill-up = (1/6 + 1/8 - 1/P) part
= (3P + 4P - 24)/24P part
= (7P - 24)/24P part

According to the question,
24P/(7P − 24) = 4

⇒ 24P = 28P − 96
⇒ 4P = 96
∴ P = 24

∴ the outgoing pipe take 24 hours to empty the tank.
In 48 hours, it can empty = 48/24 = 2 tanks

Question: Find the missing character?

Solution:
Logic:
3 × 4 + 1 = 13

13 × 4 + 1 = 53

53 × 4 + 1 = 213

so, 213 × 4 + 1 = 853

Let the length of the train be x metres and its speed be y m/s.
Then,
x/(y - a) = b and x/{y - (a + 1) = (b + 1)
⇒ x = b (y - a) and x = (b + 1)(y - a - 1)
⇒ b (y - a) = (b + 1) (y - a - 1)
⇒ by - ba = by - ba - b + y - a - 1
⇒ y = (a + b + 1).

Let the price of book = 100 tk
Price of pen = 100 + 100×25% = 125 tk
Price of penholder = 100 + 100×50% = 150 tk
Difference of penholder and pen's price = 150 -125 = 25 tk

∴ Price of the pen holder more than the price of the pen (in percentage) = {(25×100)/125}% = 20%

Total age of the members = 16 × 22 = 352 years
Total age of 15 members at least = 15 × 19 = 285 years
∴ Possible maximum age = 352 - 285 = 67 years

 Here, e^x = 5
⇒ lne^x = ln 5
⇒ xlne = ln 5 [As, lne = 1]
∴ x = 1.61 

Question: An accurate clock shows 3:00 PM. Through how many degrees will the hour hand rotate when the clock shows 9:00 PM?

Solution:
From 3.00 pm to 9.00 pm,
total time = 9 - 3 = 6 hours

We know,
Angle traced by hour hand in 12 hours =  360°

∴ Angle traced by hour hand in 6 hours, 
= (360 × 6)/12
= 180°

Question: A 42-liter mixture contains milk and water in a 3 : 4 ratio. How much milk should be added to the mixture to make the ratio equal?

Solution:
The ratio of milk to water is 3 : 4
Total portion = 3 + 4 = 7

Quantity of milk = 42 × (3/7) = 18 liters.
Quantity of water = 42 × (4/7) = 24 liters.

Let,
Quantity of milk to be added = x liters

According to the question,
(18 + x) : 24 = 1 : 1
⇒ (18 + x)/24 = 1/1
⇒ 18 + x = 24
⇒ x = 24 - 18
⇒ x = 6

∴ Quantity of milk to be added = 6 liters

Question: A 300-meter-long train passes a person in 15 seconds. The person was running at 6 km/hr in the opposite direction of the train. What is the speed of the train in km/hr?

Solution:
দেওয়া আছে,
ট্রেনের দৈর্ঘ্য = 300 মিটার
অতিক্রম করার সময় = 15 সেকেন্ড
ব্যক্তির গতিবেগ = 6 কিমি/ঘন্টা

আমরা জানি, আপেক্ষিক গতিবেগ = দূরত্ব/সময়
আপেক্ষিক গতিবেগ = 300/15 মিটার/সেকেন্ড
= 20 মিটার/সেকেন্ড
= 20 × (18/5) কিমি/ঘন্টা
= 72 কিমি/ঘন্টা

যেহেতু ট্রেন এবং ব্যক্তি বিপরীত অভিমুখে গতিশীল,
তাই আপেক্ষিক গতিবেগ = ট্রেনের গতিবেগ + ব্যক্তির গতিবেগ।

মনে করি, ট্রেনের গতিবেগ = v কিমি/ঘন্টা

প্রশ্নমতে,
v + 6 = 72
⇒ v = 72 - 6
∴ v = 66

সুতরাং, ট্রেনের গতিবেগ 66 কিমি/ঘন্টা।

Question: A sum of 12,000 Taka is lent out in two parts. One part is lent at 8% simple interest and the other at 10% simple interest. If the total annual interest is 1,080 Taka, how much is lent at 8%?

Solution:
Let, 
Amount lent at 8% simple interest = x
Amount lent at 10% simple interest = (12000 - x)

We know,
I = Pnr
I = Pr [Since the time is 1 year]

Interest from the amount lent at 8%, 
I = x(8/100) = 0.08x

Interest from the amount lent at 10%, 
I = (12000 - x) × (10/100) = 0.10(12,000−x)

ATQ,
0.08x + 0.10(12,000−x) = 1080
⇒ 0.02x = 120
∴ x = 6000 Taka

So, Amount lent at 8% simple interest = 6000 Taka

2/a + 1/b
= (2b + a)/ab
= 6/4
= 3/2

Area of the park = (60 x 40) m² = 2400 m²
Area of the lawn = 2109 m²
∴ Area of the crossroads = (2400 - 2109) m² = 291 m²

Let the width of the road be x metres. Then,
60x + 40x - x² = 291
⇒ x² - 100x + 291 = 0
⇒ (x - 97)(x - 3) = 0
⇒ x = 3 m

Let the breadth be x metre,
Then, length = 4x metre
∴ Volume of the room = (4x × 2x × x) m³
= (8x³) m³

8x³ = 512
x³ = 64
x = 4.

The length of the room is (4 × 4) = 16 m.

Question: 


Solution: 

আমরা জানি, মূলবিন্দুগামী রেখার সমীকরণ y = mx
এখানে, ঢাল m = 3
y = 3x............(i)
এখন (6, y) বিন্দুর জন্য (i) নং হতে পাই,
y = 3x
∴ y = 3 × 6 = 18 [∵ ভূজ = 6]
আবার, (x, 12) বিন্দুর জন্য (i) নং হতে পাই,
y = 3x
⇒ 12 = 3x [∵ কোটি = 12]
∴ x = 12/3 = 4
অতএব, y - x = 18 - 4 = 14

Number of shares = (4455/8.25)
= 540
Face value = Tk(540 × 10)
= Tk. 5400
Tk (12/100 × 5400) = Tk. 648

5 @ 7 
= 5 + 7 - 5 × 7
= 12 - 35
= - 23

Question: A person sold a book at 20% profit and another at 12% loss, both at Tk. 330 each. Find his overall profit or loss percent.

Solution:
Let the cost price of the first book = C1​
Let the cost price of the second book = C2​

First book:
Sold at 20% profit, Selling Price = 330
C1 × 1.2 = 330
⇒ C1 = 330 / 1.2 = 275

Second book: Sold at 12% loss,
Selling Price = 330
C2 × 0.88 = 330
⇒ C2 = 330 / 0.88 = 375

Total cost price  = 275 + 375 = 650
Total selling price = 330 + 330 = 660

Profit/Loss = SP - CP
= 660 - 650
= 10

Profit% = (10/650) × 100%
= 1.538%
= 1.54%

∴ His overall profit = 1.54%

Question: Find the midpoint of the line segment joining the points A = (- 2, 7) and B = (4, - 5).

Solution:
দেয়া আছে,
A = (- 2, 7)
B = (4, - 5)

আমরা জানি,

Question: If x and y are positive integers such that 5^{x + y} = 625 and 4^{x - y} = 16, what is the value of x² - y²?

Solution:
Given that,
5^{x + y} = 625
⇒ 5^{x + y} = 5⁴
∴ x + y = 4 ........(1)

And,
4^{x - y} = 16
⇒ 4^{x - y} = 4²
∴ x - y = 2 ....... (2)

Now (1) + (2), we get,
⇒ (x + y) + (x - y) = 4 + 2
⇒ 2x = 6
⇒ x = 6 / 2
∴ x = 3

From (1),
3 + y = 4
⇒ y = 4 - 3
∴ y = 1

∴ x² - y² = (3)² - (1)²
= 9 - 1
= 8

Face value of each share = Tk.20
Market value of each share = Tk.25
Number of shares = 12500

The amount required to purchase the shares
= 12500 × 25
= 312500